MM

Trig Graphing and Inverse Functions Notes (5.3–5.5)

  • 5.3: Graphing trig functions using a practical, step-by-step approach (different from t-tables)
    • The instructor contrasts this method with the textbook’s t-tables, aiming to sketch quickly by transforming the base graphs.
    • Core quantities to identify for a transformed sine/c cosine graph:
    • Amplitude: A = |a|, where the function is of the form y = a\, ext{sin}(b x - c) ext{ (or similar)}. In the example, the inside contains a 2, so we rewrite to handle the phase shift correctly.
    • Period: T = \frac{2\pi}{|b|}. In the example, b = 2, so T = \frac{2\pi}{2} = \pi.
    • Phase shift (horizontal shift): for y = a\sin(bx - c), the shift is \Delta x = \frac{c}{|b|} to the right (if the form uses a minus sign inside). In the discussed example, the rewriting leads to a shift discussed as either \frac{\pi}{2} to the right or, more precisely from the standard form, \Delta x = \frac{c}{b} with the given values. The teacher notes that using the form -2\sin(2x-\tfrac{\pi}{2}) fixes amplitude and period but may affect the horizontal shift depending on the representation. The takeaway is to use a consistent standard form to deduce the phase shift correctly.
    • Quick rule of thumb: rewrite inside so that you can read off amplitude, period, and phase shift cleanly; if you keep the inside as something like 2x-\tfrac{\pi}{2}, the shift is not just \tfrac{\pi}{2} unless you divide by the inside coefficient; the general rule is shift = \dfrac{c}{b} for the form \sin(bx - c).
    • Graphing plan (blueprint): start from the basic sine shape, then apply transformations in stages:
    • Step 1: apply vertical stretch/reflection due to amplitude (multiply y-values by |a|, and apply a reflection if a is negative)
    • Step 2: adjust the period by compressing/stretching the x-direction by a factor of \,\frac{1}{|b|} (i.e., horizontally scale the base graph by 1/|b|)
    • Step 3: apply the phase shift by translating the graph left/right by the shift amount
    • Example sketch notes (from the transcript):
    • Start with the default sine: looks like a regular up-and-down wave.
    • Apply amplitude 2 and reflect if negative: the graph is vertically stretched and flipped about the x-axis.
    • Apply period change (divide x-intervals by 2) to compress the wave from a 0 to 2\pi window to a smaller interval (length \pi).
    • Finally, apply the phase shift by shifting the graph to the right by the calculated amount (e.g., \pi/2 or the result c/b, depending on the chosen form).
    • Practical note: you can do all three modifications in one step with practice; otherwise, you can plot a quick sketch in stages (as demonstrated) to avoid relying on a t-table.
    • Special caveat: if you want an exact graph, you can always revert to a t-table; the method shown here is a heuristic to read off where the graph should go quickly.
    • Common pitfalls to avoid:
    • Forgetting the reflection when the amplitude is negative.
    • Miscomputing the phase shift by not accounting for the inner coefficient b.
    • Mixing up left/right direction of the phase shift.
  • Tangent and cotangent graphs (5.3 content): these are trickier because they require recognizing asymptotes and the standard shapes.
    • Tangent:
    • Asymptotes at x = \frac{\pi}{2} + k\pi (i.e., all odd multiples of \tfrac{\pi}{2}).
    • The basic graph resembles a rotated S-shaped curve with vertical asymptotes, extending infinitely in both directions within a single period.
    • With a negative amplitude, the graph is reflected across the x-axis (the shape flips).
    • For these graphs, amplitude is not used in the same way as sine/cosine graphs, because the shape is preserved up to a reflection; the key modifications are the period (via b) and the phase shift.
    • Cotangent:
    • Asymptotes at multiples of x = k\pi (i.e., 0, \pi, 2\pi, …).
    • The standard cotangent shape is the complement of tangent (sometimes described as the reflection/rotation of the tangent shape).
    • Practical note: one often needs to know the reference shapes rather than recomputing from scratch each time; if you can identify asymptotes and a central point, you can sketch quickly.
  • Secant and cosecant graphs (5.3 content): how they relate to sine and cosine.
    • Secant ( sec x ) graphs:
    • Asymptotes where cos x = 0 (i.e., at x = \tfrac{\pi}{2} + k\pi).
    • Graphs appear as two parabolic-like branches opening upward or downward depending on the sign of cos x, with the branches located near where cos x is near ±1.
    • They’re not simply a single parabola; the secant graph consists of two mirrored branches in each period.
    • Cosecant ( csc x ) graphs:
    • Asymptotes where sin x = 0 (i.e., at x = k\pi).
    • The graph consists of two parabolic-like branches occurring above and below the x-axis.
    • Important practical takeaway: for both secant and cosecant, you derive the graph by starting from the sine/cosine base graphs and applying reciprocal relationships; the asymptotes come from zeros of the underlying sine/cosine.
  • Exam emphasis (5.3/5.4 shift): focus tends to be on sine, cosine, and tangent; secant and cosecant appear less frequently on exams, though they’re part of the same transformation toolkit.
    • The instructor notes that in exams they try to limit to the big three (sine, cosine, tangent) most of the time, with occasional inclusion of secant/cosecant on quizzes or homework.
  • 5.5: Inverse Trigonometric Functions
    • Core idea: to find an inverse, you swap x and y and solve for y, but the inverse must be a function, which requires the original function to pass the horizontal line test (one-to-one).
    • Separation of tests:
    • Vertical line test determines if f is a function (usual). Horizontal line test determines if the inverse is a function (i.e., f must be one-to-one).
    • For common trig functions, sine, cosine, and tangent, none of them are one-to-one on their entire usual domains, so their inverses are not functions unless restricted to a suitable domain.
    • Domain/range restrictions for principal inverses (to make them functions):
    • Sine inverse: ext{arsin}(x) or \sin^{-1}(x)
      • Domain: x \in [-1, 1]
      • Range: y \in [-\tfrac{\pi}{2}, \tfrac{\pi}{2}]
    • Cosine inverse: \cos^{-1}(x)
      • Domain: x \in [-1, 1]
      • Range: y \in [0, \pi]
    • Tangent inverse: \tan^{-1}(x)
      • Domain: x \in \mathbb{R}
      • Range: y \in (-\tfrac{\pi}{2}, \tfrac{\pi}{2})
    • Practical consequences:
    • If you see sin^{-1}(π) or cos^{-1}(2), there is no solution because the input to the inverse must lie in the function’s domain ([-1,1] for sin/cos inverses).
    • If you see an expression like sin^{-1}(sin θ) and θ is not within the principal range, you cannot simply cancel; you must map θ back into the principal interval to find the corresponding inverse value.
    • Arc notation synonyms: arc sine = sin^{-1}, arc cosine = cos^{-1}. The term arc is widely used to denote the inverse function; some instructors prefer to write arcsec, arccsc, arccot (arcsec, arccsc, arccot are the inverse functions for sec, csc, cot respectively).
    • Canceling rules (inside vs outside inverses):
    • If the inverse is inside the argument (e.g., sin( arcsin(x) )), it cancels straightforwardly: sin(arcsin(x)) = x (for x in the domain) and generally the inside cos or sin cancels when the inner function is the exact inverse of the outer function and the input lies in the corresponding range.
    • If the inverse is outside (e.g., arcsin( sin(x) )), you cannot simply replace with x; you must consider the restricted range of the inverse and use domain-constrained angles (principal values).
    • Examples and conventions discussed:
    • If you have arc cosine of cosine of an angle, you can cancel when the angle is within the principal value [0, π]. For example, \cos^{-1}(\cos \theta) = \theta if \theta \in [0, \pi]. Outside this interval, the result is the equivalent angle within the principal range rather than the original angle.
    • Sine and cosine inverses require attention to the principal domain/range; tangent inverse is generally easier to handle because its range is the open interval (-π/2, π/2).
    • Using reciprocals to handle inverse secant/cosecant/cotangent:
    • Since secant is the reciprocal of cosine, arcsec(x) can be treated as arccos(1/x) (with attention to the domain/range of arcsec). Similarly, arccsc(x) can be treated as arcsin(1/x).
    • For cotangent, a direct inverse is less common in calculators; reciprocals with tangent can be used, but you must respect the corresponding domain/range.
    • The transcript illustrates a practical trick: if you need arcsec(x) or arccsc(x), convert to the corresponding arc cosine or arc sine using the reciprocal identity first, then apply the inverse on the simpler function.
    • A concrete example from the transcript:
    • Compute the inner-to-outer chain where the inverse is inside a composition, e.g., arc cosine of cosine of an angle within [0, π] will cancel to the angle; but outside that interval, you must map back to the principal value.
    • A common composite example given: write secant of tangent inverse of x as a function of x.
    • Let u = \tan^{-1}(x). Then the expression becomes \sec(u) with \tan(u) = x.
    • Use the identity \sec^2(u) = \tan^2(u) + 1, hence \sec(u) = \pm \sqrt{\tan^2(u) + 1} = \pm \sqrt{x^2 + 1}.
    • The ± arises because the quadrant of u is not specified; without quadrant information, both signs are possible. If a quadrant is specified, you choose the sign accordingly.
    • Quick practical tips: keep the order of operations in inverse problems; when the inner function is an inverse, you can often cancel; when the inverse is outside, work step-by-step through the inside before applying the outer inverse.
  • Quadrant-based heuristics (helpful for inverses):
    • For sine inverse, the result must lie in [-\tfrac{\pi}{2}, \tfrac{\pi}{2}]. Therefore, when you’re given a sine value and asked for the angle, locate the corresponding quadrant(s) that produce that sine value within the principal range.
    • For cosine inverse, the result must lie in [0, \pi]. Positive cosine values correspond to Quadrant I or IV, but within the principal range you select Quadrant I if the value is positive and Quadrant II if negative.
    • For tangent inverse, the result must lie in (-\tfrac{\pi}{2}, \tfrac{\pi}{2}); signs and quadrants are chosen accordingly based on the sign of the input.
    • A practical mnemonic from the transcript: when the inverse is “inside” (i.e., the inverse function is inside the argument of another function), cancelation is often straightforward; when the inverse is “outside,” work through the inside first.
  • Worked example-oriented notes (summary of strategies):
    • When encountering a problem like arccos(cos(θ)) and θ is within the principal value range, you can cancel and get θ; if θ is outside, you adjust to the corresponding angle within the principal range.
    • For expressions involving mixed inverses and non-inverse trig functions, follow the inside-out approach and use fundamental identities (Pythagorean and reciprocal identities) to simplify.
    • Be mindful of the domain restrictions: inputs to inverse trig functions must lie in the function’s domain; if not, there is no solution.
  • Quick study takeaways:
    • Know the principal value ranges for inverse trig functions: arcsin: [-\tfrac{\pi}{2}, \tfrac{\pi}{2}], arccos: [0, \pi], arctan: (-\tfrac{\pi}{2}, \tfrac{\pi}{2}).
    • For inverse problems, always check domain before applying inverse functions; otherwise you may encounter domain errors or no-solution scenarios.
    • Use reciprocal identities to handle inverse secant/cosecant/cotangent when calculators do not provide those inverses directly.
    • The exam tends to emphasize the three main functions (sine, cosine, tangent) for inverse questions, with additional care given to domains and principal values; less emphasis on secant/cosecant on typical exams.
  • Final synthesis (exam-oriented):
    • Expect to handle both direct trig graphs with standard transformations and inverse trig problems with domain restrictions and principal values.
    • Build fluency by: (i) recognizing when you can cancel inverses (inside) and when you must work through the inside step-by-step (outside); (ii) applying core identities to convert between functions when needed; (iii) being comfortable with the principal value ranges and quadrant reasoning to locate correct inverse values.
    • The instructor emphasizes using unit-circle familiarity, recognizing asymptotes and standard shapes, and practicing with a mix of sine, cosine, tangent problems (with occasional secant/cosecant considerations) to prepare for exams.